For a general closed subset \( K \)  it is well known that \(\delta _{K}^{\phi } \)  is a
locally semiconcave function on \( \mathbf {R}^{n} \operatorname{\sim }K \) ; see [1]},
[2]}, [3]}. This implies that its gradient \( \operatorname{grad}\delta _{K}^{\phi } \)  is a function of bounded variation, see [4]}
(see also [5]} for further results in this direction). Employing general results from the theory of Hamilton-Jacobi
equations (see [2]} or [7]}), one can also
obtain the following \( \mathcal {C}^{1,1} \) -regularity theorem:
